A pair of sequences (αn(a,k,q),βn(a,k,q)) such that α0(a,k,q)=1 and βn(a,k,q)=∑nj=0 (k/a;q)n−j (k;q)n+j / (q;q)n−j (aq;q)n+j αj (a,k,q) is termed a WP-Bailey Pair. Upon setting k=0 in such a pair we obtain a Bailey pair.
In the present paper we consider the problem of “lifting” a Bailey pair to a WP-Bailey pair, and use some of the new WP-Bailey pairs found in this way to derive some new identities between basic hypergeometric series and new single-sum and double-sum identities of the Rogers–Ramanujan–Slater type.
McLaughlin, James, Andrew Sills, Peter Zimmer.
"Lifting Bailey Pairs to WP-Bailey Pairs."
Discrete Mathematics, 309 (16): 5077-5091.
doi: 10.1016/j.disc.2009.03.015 source: http://www.math.rutgers.edu/~asills/WPBailey/WPBailey.pdf